Signaling Games in Multiple Dimensions: Geometric Properties of Equilibrium Solutions

Abstract

Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. We investigate a signaling game problem where an encoder observes a multi-dimensional source and conveys a message to a decoder, and the quadratic objectives of the encoder and decoder are misaligned due to a bias vector. We first provide a set of geometry conditions that needs to be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and completely characterize conditions under which an informative linear Nash equilibrium exists. In particular, we show that if the components of the bias vector are not equal in magnitude, then there exists a linear equilibrium if and only if the source distribution is Gaussian. On the other hand, for a linear equilibrium to exist in the case of equal bias components, it is required that the source density is symmetric around its mean. Moreover, in the case of Gaussian sources, our results have a rate-distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from their team theoretic counterpart.

Quadratic Privacy-Signaling Games and the MMSE Information Bottleneck Problem for Gaussian Sources

“…In fact, depending on whether the 542 source is scalar or vector valued, there may exist linear 543 informative Nash equilibria for the signaling game setup with 544 a biased sender. In particular, in [38], we extend Crawford 545 and Sobel's formulation to a multidimensional source setting 546 under quadratic cost criteria with a biased sender and show 547 that for independent and identically distributed sources and an 548 arbitrary bias vector, there always exist linear informative Nash 549 equilibria (only) when the source distribution is Gaussian.…”

“…The decoding policies at this equilibrium are given by the minimum mean squared error estimators corresponding to each random variable. Since the observation R is jointly Gaussian with X and Y , the conditional expectation formula for Gaussian distributions can be employed to obtain (37) and (38) [63, p. 155].…”